8.2 Some formal considerations

Before we proceed with a more involved example, let us formalize the procedure outlined above. We
mainly follow the excellent treatment given in [124], although we restrict ourselves to the cases where is
a finite regular subalgebra of .

In the previous example, we performed the decomposition of the roots (and the ensuing decomposition
of the algebra) with respect to one of the simple roots which then defined the level. In general, one may
consider a similar decomposition of the roots of a rank Kac–Moody algebra with respect to an arbitrary
number of the simple roots and then the level is generalized to the “multilevel”
.

8.2.1 Gradation

We consider a Kac–Moody algebra of rank and we let be a finite regular rank
subalgebra of whose Dynkin diagram is obtained by deleting a set of nodes
from the Dynkin diagram of .

Let be a root of ,

To this decomposition of the roots corresponds a decomposition of the algebra, which is called a gradation
of and which can be written formally as

where for a given , is the subspace spanned by all the vectors with that definite value of
the multilevel,

Of course, if is finite-dimensional this sum terminates for some finite level, as in Equation (8.11) for
. However, in the following we shall mainly be interested in cases where Equation (8.13) is an
infinite sum.

We note for further reference that the following structure is inherited from the gradation:

This implies that for we have

which means that is a representation of under the adjoint action. Furthermore, is a
subalgebra. Now, the algebra is a subalgebra of and hence we also have

so that the subspacesat definite values of the multilevel are invariant subspaces under theadjoint action of. In other words, the action of on does not change the coefficients
.

At level zero, , the representation of the subalgebra in the subspace contains
the adjoint representation of , just as in the case of discussed in Section 8.1. All positive and
negative roots of are relevant. Level zero contains in addition singlets for each of the Cartan
generator associated to the set .

Whenever one of the ’s is positive, all the other ones must be non-negative for the subspace to
be nontrivial and only positive roots appear at that value of the multilevel.

8.2.2 Weights of and weights of

Let be the module of a representation of and be one of the weights occurring in
the representation. We define the action of in the representation on as

(we consider representations of for which one can speak of “weights” [116]). Any representation of
is also a representation of . When restricted to the Cartan subalgebra of , defines a weight
, which one can realize geometrically as follows.

The dual space may be viewed as the -dimensional subspace of spanned by the simple
roots , . The metric induced on that subspace is positive definite since is finite-dimensional.
This implies, since we assume that the metric on is nondegenerate, that can be decomposed as
the direct sum

To that decomposition corresponds the decomposition

of any weight, where and . Now, let (). One has
because : The component perpendicular to drops out.
Indeed, for .

It follows that one can identify the weight with the orthogonal projection of
on . This is true, in particular, for the fundamental weights . The fundamental weights
project on for and project on the fundamental weights of the subalgebra for
. These are also denoted . For a general weight, one has

and

The coefficients can easily be extracted by taking the scalar product with the simple roots,

a formula that reduces to

in the simply-laced case. Note that even when is non-spacelike.

8.2.3 Outer multiplicity

There is an interesting relationship between root multiplicities in the Kac–Moody algebra and weight
multiplicites of the corresponding -weights, which we will explore here.

For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real
roots of indefinite Kac–Moody algebras. However, as pointed out in Section 4, the imaginary
roots can have arbitrarily large multiplicity. This must therefore be taken into account in the
sum (8.13).

Let be a root of . There are two important ingredients:

The multiplicity of each at level as a root of .

The multiplicity of the corresponding weight at level as a weight in
the representation of . (Note that two distinct roots at the same level project on two
distinct -weights, so that given the -weight and the level, one can reconstruct the root.)

It follows that the root multiplicity of is given as a sum over its multiplicities as a weight in the various
representations at level . Some representations can appear more than once at
each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outermultiplicity, which counts the number of times each representation appears at level . So,
for each representation at level we must count the individual weight multiplicities in that representation
and also the number of times this representation occurs. The total multiplicity of can then be written
as

This simple formula might provide useful information on which representations of are allowed within
at a given level. For example, if is a real root of , then it has multiplicity one. This
means that in the formula (8.25), only the representations of for which has weight
multiplicity equal to one are permitted. The others have . Furthermore, only one of the
permitted representations does actually occur and it has necessarily outer multiplicity equal to one,
.

The subspaces can now be written explicitly as

where denotes the module of the representation and is the number of
inequivalent representations at level . It is understood that if for some and ,
then is absent from the sum. Note that the superscript labels multiple modules
associated to the same representation, e.g., if this contributes to the sum with a term

Finally, we mention that the multiplicity of a root can be computed recursively
using the Peterson recursion relation, defined as [116]

where denotes the set of all positive integer linear combinations of the simple roots, i.e., the positive
part of the root lattice, and is the Weyl vector (defined in Section 4). The coefficients are defined
as

and, following [19], we call this the co-multiplicity. Note that if is not a root, this gives no
contribution to the co-multiplicity. Another feature of the co-multiplicity is that even if the
multiplicity of some root is zero, the associated co-multiplicity does not necessarily
vanish. Taking advantage of the fact that all real roots have multiplicity one it is possible,
in principle, to compute recursively the multiplicity of any imaginary root. Since no closed
formula exists for the outer multiplicity , one must take a detour via the Peterson relation and
Equation (8.25) in order to find the outer multiplicity of each representation at a given level. We give
in Table 38 a list of root multiplicities and co-multiplicities of roots of up to height
10.